Dielectric Properties of Water at Rutile and Graphite Surfaces: Effectof Molecular StructureStanislav Parez,*,† Milan Predota,†,‡ and Michael Machesky§

†Institute of Chemical Process Fundamentals, Academy of Sciences of the Czech Republic, Prague, 165 02, Czech Republic‡Faculty of Science, University of South Bohemia, Branisovska 1760, Ceske Budejovice, 370 05, Czech Republic§University of Illinois, State Water Survey, 2204 Griffith Drive, Champaign, Illinois 61820, United States

ABSTRACT: The dielectric properties of interfacial wateradjacent to the surfaces of hydrophobic graphite and the 110surface of hydrophilic rutile (α-TiO2) are investigated bymeans of nonequilibrium molecular dynamics simulations. Thedielectric behavior of water is found to arise from its localdensity and molecular polarizability in response to an externalfield, and can be rationalized in terms of the number andstrength of water−surface and water−water H-bonds. Theinterplay of local density and polarizability leads to aparticularly strong dielectric response, exceeding the externalfield, of the water layer directly contacting the surfaces, whilethe second layer exhibits a reduced response. Consequently, dielectric profiles near surfaces cannot be correctly described byimplicit solvent models valid for bulk water. The overscreening response of the contact water layer has been observed in previoussimulation studies and implies the local permittivity (dielectric constant) of that layer is negative. However, the negativepermittivity of the contact water layer is counterbalanced by the positive permittivities of the surface depletion layer and thesecond water layer such that the calculated Stern layer capacitance is positive and compatible with experimental data. Moreover,the electrostatic potential profile matches well the profile calculated for an aqueous salt solution at the charged rutile (110)surface, thus supporting the “water centric” view of aqueous electrical double layers.

1. INTRODUCTION

Molecular dynamics (MD) simulations can provide valuableinformation about the structure and dynamics of water, ions,and molecules at the solid−liquid interface. Mineral surfacescontacting pure water only have been the subject of mostprevious MD simulations and include titanium dioxide,1,2 clay,3

carbonate,4 and sulfide5 minerals. Simulations in which mineralsurfaces are in contact with both water and ions are rarer butinclude clay minerals,6,7 cassiterite (SnO2),

8 and gibbsite.9

The (110) rutile (α-TiO2) surface in contact with aqueoussolutions has been studied by MD simulation in our previouspapers. The interfacial structure of water and ions from thosesimulations agrees very well with experimental data obtained byX-ray standing wave and X-ray reflectivity (e.g., crystaltruncation rod) analysis of surface structures at ambientconditions,10−14 which can be treated as validation of oursimulations and enables us to extend the range of investigatedproperties. In the ongoing research, we have studieddiffusivity,15 shear viscosity,16 and hydrogen-bonding struc-ture17,16 of the rutile−water interfacial region. The resultsdemonstrate that interfacial properties can differ significantlyfrom bulk behavior, which in turn questions continuumtheories in which bulk values or bulk dependences are usedin place of local counterparts.This Article continues our investigation of rutile−aqueous

solution interfacial phenomena by addressing dielectric proper-

ties. Our goal is to characterize the relative permittivity(dielectric constant) profile in the inhomogeneous regionformed at the solid−liquid interface, as it changes from bulkvalues far from the surface to the vicinity of the surface.Moreover, our aim was not to calculate the “dielectric function”with a submolecular resolution, as in recent simulationstudies,18 but to determine the permittivity of individualmolecular layers of water at the surface. The reasons are 2-fold.First, permittivity is a quantity established in continuumtheories of dielectrics to approximate electrostatic properties ofa macroscopic volume in a polarizable solvent. We consider thisconcept reasonable on the scale exceeding or at least equal to amolecular length. Second, local permittivity can be analyzed interms of other structural properties of the molecular layers. Tothat end, we will provide a molecular level explanation for thepermittivity profile of water at the (strongly hydrophilic) rutile(110) surface, as well as the hydrophobic graphite surface forcomparative purposes.Direct measurements of interfacial permittivity are difficult

due to the small thickness of the region over which permittivityvaries. Dielectric spectroscopy provides dielectric relaxationspectra over a range of frequencies. Despite the signal

Received: December 31, 2013Revised: February 11, 2014Published: February 12, 2014

Article

pubs.acs.org/JPCC

© 2014 American Chemical Society 4818 dx.doi.org/10.1021/jp4128012 | J. Phys. Chem. C 2014, 118, 4818−4834

originating both from interfacial and from bulk environments,spectra can be decomposed into components corresponding toprocesses with different relaxation times, and the contributionof interfacial water can be assigned.19,20 However, the inferredvalue of interfacial permittivity depends on the properinterpretation of experimental spectra. Recently, terahertzspectroscopy has been introduced as a tool for directly probinginterfacial water dynamics, and experiments on carbohydrates21

and lipid membranes22 have shown that the dielectric responseof water is modified within a few solvation layers. Atomic forcemicroscopy for water at mica surfaces was applied by Teschkeet al.,23 and their results showed a slow decrease in permittivityextending over ∼25 nm as the surface was approached. Incontrast, spectroscopic measurements (e.g., ref 21) andmolecular simulations (e.g., ref 18) show permittivity variesonly over a distance of a few molecular lengths. Bernik andNegri24,25 studied the local polarity of phospholipid vesicles byanalyzing the Stokes shift of fluorescent probes. They observedanomalous values of permittivity, such as negative or extremelylarge values around 100, at the phospholipid−water interface.However, the anomalous values were interpreted as a spuriousresult due to uncertainties in estimating the local refractiveindex.Obstacles to the detailed measurement of interfacial

permittivity have resulted in the development of theoreticalapproaches and molecular simulations. Dielectric properties arefrequently discussed in the framework of the electrical doublelayer (EDL) and its various constructs.26−29 A widely usedconstruct is the Stern (or Gouy−Chapman−Stern) model,30,31which partitions an interface into two layers: (i) the Stern layer(or Helmholtz layer), referring to the compact layer ofimmobile ions adsorbed at the surface, and (ii) the diffuselayer where the ions are mobile and their concentration profileobeys the Poisson−Boltzmann equation.32,33 The role of wateris that of a continuum with relative permittivity εr,i filling thelayer i. The model is parametrized with a Stern layercapacitance value, which is commonly fit to experimentalsurface titration data.27,29,34 For metal oxide surfaces in contactwith 1:1 electrolyte solutions, Stern layer capacitance values canalso be predicted with the empirical relationships developed bySverjensky,35,36 which are strictly valid for the triple layer (twoStern layers and a diffuse layer) surface complexation model.Capacitance values were predicted to vary with both electrolytetype and oxide surface considered, with capacitance values forthe innermost Stern layer ranging from about 0.5 to 1.6 F/m2.Values of permittivity do not follow directly from modeled

capacitances, but can be linked to information on the positionsof the Stern layers from X-ray reflectivity and other techniquesaccording to the equation Ci = εr,iε0/di, where Ci and di are thecapacitance and thickness of layer i, respectively. Stern layerpermittivity values derived from EDL models vary between ∼6for metal electrodes to 20−60 for metal oxides.29,35 However,because permittivity values are sensitive to the particular modelapplied,27,29,34 knowledge of interfacial permittivity via othermethods would enable the assessment of various EDL models.Classical EDL models are examples of “implicit solvent

models”, which describe electrostatic profiles of water on a basisof mean field approximations (while the electrostatic potentialarising from ions is often accounted for explicitly). Specifically,in the Poisson−Boltzmann equation used for the calculation ofthe total electrostatic profile, electrostatic potential is related toan explicit ionic charge distribution (assuming pointwise orfinite size ions), whereas the electrostatic potential generated by

water is accounted for by a prescribed permittivity profile.Several formulas for calculating permittivity profiles of watercan be found in the literature, from a simple constant bulk valueto distance- or field-dependent permittivity.31,37−39 However,these formulas, for example, permittivity variance with electricfield, were derived for homogeneous bulk systems, and theseimplicit solvent models completely ignore the effect ofinterfacial interactions on dielectric properties. The main goalof this Article is to show that the interfacial solvent and bondnetwork structure indeed affect electrostatics, and to comparethe effect to the behavior in homogeneous solution.Simulation studies of interfacial dielectric properties have

primarily addressed pure water at neutral surfaces, and differ inthe simplicity of the molecular models used. For the simplestcase of a smooth dielectric wall, dielectric profiles can becalculated semianalytically.40 More complex surface modelsinvolve silica41,42 or membrane bilayers.43−45 A system similarto the present one, the quartz−water interface, was examinedunder different hydration levels by Wander and Clark.46 Theyfound permittivity to monotonously decrease toward thesurface in agreement with the EDL picture, for example, ref36. However, they calculated permittivity using the Kirkwoodfluctuation formula,47,48 where correlations between individualwater dipoles at zero external field appear. The plausibility ofusing this approach for interfaces is questionable at best. First,this approach is strictly applicable to homogeneous isotropicbulk fluids assuming the studied system is symmetricallysurrounded by a dielectric medium.47 Second, when applied toa water layer within the interfacial region, correlations withdipoles outside the layer are neglected, although theircontribution might be substantial if water is strongly ordered.Third, contributions from dipole moment fluctuations due tomolecules entering/leaving a particular layer are not properlyaddressed. These drawbacks were addressed by Ballenegger andHansen,49 who derived a correct fluctuation formula applicableto planar interfaces and calculated the permittivity (tensor) fora simple model of a dipolar liquid at a homogeneous dielectricwall. Bonthuis et al.18,50 generalized the fluctuation formula forthe case of fluid molecules containing higher multipolemoments and applied the formula to more realistic SPC/Ewater adjacent to planar, atomistically modeled, hydrophilic andhydrophobic diamond surfaces. Application of the correctfluctuation formula revealed a novel feature in the electrostaticprofile: the local electric field in the contact water layer had asign opposite to that of the external field.49,50 This “over-screening” of the external field by the dielectric responseultimately results in a negative value of permittivity (applied tothe transverse component, i.e., perpendicular to the interface).Bonthuis et al. employed the permittivity profile in thePoisson−Boltzmann equation and succeeded in roughlymatching experimental double layer capacitances as a functionof electrolyte concentration.18,50 However, the comparison wasdone against experimental capacitance data from variouscarbon-based surfaces (including diamond) that were highlyvariable with respect to electrolyte concentration. In addition,calculated capacitance values and their concentration depend-ence were highly dependent on the definition of the surfacecharge position.50 Still, calculated capacitances were similar forhydroxylated (hydrophilic) and nonhydroxylated (hydropho-bic) diamond surfaces, and it was concluded that thecorresponding electrostatic profiles were dominated by waterrather than ionic contributions. We will confirm the latterconclusion based on explicit calculations of electrostatic profiles

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp4128012 | J. Phys. Chem. C 2014, 118, 4818−48344819

from simulations of aqueous salt solutions at charged rutilesurfaces.Previous simulation studies on interfacial permittivity profiles

have utilized pure solvent simulations, thereby neglecting theeffect of ions on permittivity. We will examine simulations of acharged rutile surface in contact with NaCl electrolyte todetermine whether the electrostatic profile reflects primarilyionic or intrinsic water properties by calculating the ionic andwater contributions to the electrostatic profile explicitly fromthe simulation. However, determination of permittivityexplicitly from such simulations is beyond the scope of thisArticle. For pure solvent systems (without free charges), theconstitutive relation defining permittivity between displacementD and electric E fields is local:51 D(r) = ε0εr(r)E(r), while ifions are included permittivity might show a nonlocaldependence.18,52 Generally, local permittivity is a goodapproximation when the displacement field varies slowly ascompared to the electric field, that is, at low salt concentration.A simulation study of a realistic silica surface42 revealednonlocal behavior of the dielectric function over the Stern layer,while the typical local permittivity description was adequate inthe diffuse layer.In this Article, we investigate the effect of water structure and

local bonding on interfacial permittivity for two surfaces with avastly different influence on the adjacent water: the stronglyhydrophilic rutile (110) surface and a hydrophobic graphitesurface. We first review the expressions used for calculatingpermittivity and use those expressions to determine permittivityin the molecular layers of interfacial water. Second, we give amolecular level explanation for the observed permittivity values.Local permittivity is related to density of a water layer and anorientational response of its molecules to the external field withthe orientational response rationalized in terms of the local H-bonding structure. Third, values for the rutile surface are usedto calculate the Stern layer capacitance and are compared toresults from surface complexation modeling. In addition, theelectrostatic profile from the simulation of a charged rutilesurface is analyzed to determine the separate contributions ofions and water. Finally, we discuss the role of interfacialmolecular structure for dielectric properties and the corre-sponding validity of implicit solvent models where structure isneglected.

2. METHODOLOGY

When a sample of a polar fluid is exposed to an external fieldEext, it becomes polarized and produces a polarization field Ep,which acts typically opposite to the external field. Let usconsider a fluid confined in a slab under an externalhomogeneous electric field acting in the direction normal tothe surfaces. The planar symmetry (neglecting effects due toroughness of the surfaces) ensures that the polarization field isnonzero only in its z-component, which is normal to thesurface, Ep Ep

z. Its value can be derived from chargedistribution or from multipole expansion. The chargedistribution approach requires knowledge of the microscopiccharge distribution ρq, such as the locations and magnitudes ofpartial charges that appear in Gauss’s law:51

∫ερ= ′ ′

−∞E z z z( )

1( ) d

z

qp0 (1)

where ε0 is the permittivity of a vacuum and the coordinate zmeasures distance from a surface. In the multipole expansion

approach, the polarization field is generated by the polarizationdensity vector P:18,51

ε= −z

zE

P( )

( )p

0 (2)

which comprises individual multipole density Pi contribu-tions:18

= − ∇· + ∇∇ −z z z zP P P P( ) ( ) ( ) : ( ) ...1 2 3 (3)

The leading term of the expansion is the dipole momentdensity P1. The equivalency of the charge density approach, eq1, and the multipole expansion approach, eq 2, is ensured bythe following relation:

∫ ρ= − ′ ′−∞

P z z z( ) ( ) dz

q (4)

where P Pz.The dielectric response of the sample (we will use the term

“dielectric response” or “response” for the change in thepolarization field ΔEp on applying the external field; in abroader context, “response” can also refer to changes inmicroscopic structure of the sample driving the change in thepolarization field) is often expressed via the relative permittivityεr. Assuming the local relation between polarization density andelectric field, permittivity is defined through the linearconstitutive relation:

ε ε ε= +E E Pr 0 0 (5)

where E is the total electric field, that is, the sum of the externalfield and the polarization field:

= +E E Eext p (6)

According to eqs 2 and 5, polarization density is a linearfunction of the external electric field. This applies for smallfields unless the polarization becomes saturated, whereby nearlyall molecules are already aligned with the field and cannot reactfurther by changing their orientation. For higher fields causing anonlinear response, permittivity is allowed to be a function ofthe field to preserve the linear constitutive relation eq 5.The definition of permittivity is more subtle when a system

with an interface is considered, because a wall-inducedpolarization might exist even at a vanishing external field. Asa result, polarization is no longer proportional to the externalfield. To describe the response of the fluid to the changesinduced by the external field, the values of the polarizationdensity and electric field in eq 5 are replaced by their changesΔP and ΔE with respect to the reference values at zero externalfield:

ε ε εΔ = Δ + ΔE E Pr 0 0 (7)

By construction, ΔP as well as ΔE = Eext + ΔEp vanish for avanishing external field. The modified constitutive relation eq 7thus enables comparison of permittivity in the bulk and at aninterface for the same Eext. Note that outside the interfacialregion the wall-induced polarization is zero as the bulkstructure is recovered, and eq 7 reduces to the standardexpression eq 5.Considering the relation eq 2 between polarization and

electric field, eq 7 leads to the following expression forpermittivity:

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp4128012 | J. Phys. Chem. C 2014, 118, 4818−48344820

ε =Δ

=+ Δ

=−

zEE z

EE E z SF z

( )( ) ( )

11 ( )r

ext ext

ext p (8)

where we have introduced the screening factor SF −ΔEp/Eext, which is a dimensionless factor describing that portion ofthe external field that is screened by the polarization field. Inparticular, SF = 0 if ΔE = Eext (no screening of the externalfield, εr = 1), while SF = 1 if ΔE = 0 (complete screening of theexternal field, εr = ∞).Note that eq 8 is valid for the external electric field acting

perpendicularly to the walls. Unlike in the bulk wherelongitudinal permittivity εr

∥ = 1 + Px/ε0Eext derived for theelectric field that acts parallel to surfaces yields the same value(ref 53 for water), permittivity in the interfacial region isanisotropic18,49 due to fluid−surface interaction.Permittivity can also be calculated on the basis of linear

response theory47 from molecular fluctuations of the systemwithout the applied external field. This approach leads forhomogeneous isotropic fluids to the well-known Kirkwoodfluctuation formula:47−49

ε εε ε

− += ⟨ ⟩ − ⟨ ⟩

Vk TM M

( 1)(2 1) 1( )r r

r 0 B

20 0

2

(9)

where kB denotes the Boltzmann constant, T is the absolutetemperature, M is the total dipole moment of molecules in thevolume V, and ⟨...⟩0 denotes the ensemble average for zeroexternal field. For slab geometry, the correct fluctuation formulais18,49

εε

= − ⟨ ⟩ − ⟨ ⟩ ⟨ ⟩− zk T

P z M P z M( ) 11

( ( ) ( ) )r1

0 B0 0 0

(10)

where M Mz = ∫ P(z) dV.

3. SIMULATIONSMD simulations of water slabs for a range of externalhomogeneous electric fields in the z direction were performedusing our own parallel simulation code.The TIP4P/2005 molecular model of water by Abascal and

Vega54 was employed. It consists of one Lennard-Jones (LJ)site and three point charges. This model is one of the mostsuccessful among the rigid nonpolarizable water models withrespect to various structural and transport properties.55 Usingthe molecular model, charge density and the dipole momentdensity vector were calculated bin-wise as

∑ ∑ ∑ρ = =α

αα

α αzV

q zV

qP r( )1

; ( )1

qi V

ii V

i ib { , }in

, 1b in

, ,

b b

(11)

where Vb is the volume of the parallel bin out to distance z fromthe surface, and qi,α and ri,α are the charge, and position vectorof site α at water molecule i, respectively. In the first equation,the summation runs over all sites within Vb, while in the latterequation, only water molecules with their center of mass(COM) inside Vb contribute.To see the effect of different water−wall interactions, two

different types of walls were considered, a graphite wall and arutile (110) wall. The interaction between water and thegraphite wall was described by the LJ 9-3 potential:

ρ σσ σ

= πε −⎡⎣⎢

⎤⎦⎥U z

z z( ) 4

( / )45

( / )6COM w

3 COM9

COM3

(12)

where zCOM is the distance of the water molecule center of massfrom the wall. The values of interaction parameters σ = 3.28 Åand ε/kB = 51.08 K and the number density of carbon atoms ρw= 112.9125 nm−3 were taken from ref 56. The rutile (110) wall,detailed in our previous works,10,57 represents a more realisticsurface. This all-atom model reflects the structure of neutralnonhydroxylated rutile (110) surface.10 Wall atom positionswere held rigid because no significant effects of surfaceflexibility were expected. The rutile wall consisted of fourTiO layers, where the two deepest maintained the strictlyperiodic bulk crystal structure, while the other two were abinitio relaxed.57 The surface is terminated by rows of bridgingoxygens that protrude out from the 110 surface plane. Theinteraction parameters between the rutile wall and water arebased on the ab initio-derived potentials.10,57 These potentialsdescribe Ti−O(water) interactions in terms of Buckinghampotentials, while the O(wall)−O(water) interaction is given bythe LJ potential.The applied external electric field Eext values were ±0.05,

±0.1, ±0.4, ±0.5, ±0.9, ±1, ±1.4, and ±1.5 V/Å. The weakestfield applied was ±0.1 V/Å for the graphite wall and ±0.05 V/Åfor the rutile wall. Weaker applied fields resulted in polarizationchanges from zero-external-field values comparable to statisticaluncertainty, and hence of poor accuracy. The results for ±Eextwere obtained from a single simulation, because in each casethe external field was oriented parallel to the surface normal ofone surface, but antiparallel to the surface normal of theopposite surface. A positive field is taken to point toward thefluid phase (see Figure 1).Simulations were carried out for 3000 water molecules. The

unit simulation cell was a rectangular prism with a base parallelto the surface of area 35.5 × 39.0 Å2. These dimensions werechosen to match 12 × 6 replicas of the rutile (110) unit cell.The separation of the two opposing walls in the z direction wasadjusted so that the density of water in the bulk region was 1.0g/cm3. The number of water molecules, density, and unit celldimensions resulted in a slab thickness L of almost 70 Å.Because the inhomogeneous interfacial region extends typicallyto about 15 Å from the surface, the bulk phase length wasnearly 40 Å.Periodic boundary conditions were applied on the unit

simulation cell in all three directions, and the three-dimensionalEwald summation technique with the dipole correction term(EW3DC) by Yeh and Berkowitz58 was employed to treat long-range Coulombic interactions. This method yields an excellentagreement with the results from the rigorous two-dimensionalEwald summation technique47,58 while reducing computingtime significantly. For optimal performance, a vacuum space ofthickness 50−70 Å separated the nearest images in the zdirection. The cutoff radius was set to 13 Å for both the LJinteraction and the real space Ewald summation.Simulations were equilibrated over 106 time steps, followed

by ∼6 × 106 time steps for production runs. The fifth-orderGear predictor-corrector integrator47 was used to propagate thetrajectories with a 0.9 fs time step. The temperature wascontrolled with the Nose−Hoover thermostat.47,59,60 Predictedvalue statistical uncertainties were estimated with the blockaveraging method.61

4. RESULTS4.1. Dielectric Properties of Molecular Layers. MD

simulations of water confined between parallel rutile (110) andgraphite walls were conducted at 298 K and 1 g/cm3 density. A

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp4128012 | J. Phys. Chem. C 2014, 118, 4818−48344821

strong external electric field was applied in the direction normalto the surfaces, and the response of the system was monitored.Particular attention was paid to the polarization changes ofinterfacial water layers to estimate their relative permittivities.Interfacial polarization has its origin in a strong layering and

preferential orientation of water in the vicinity of a surface, asshown in Figure 1. In Figure 2, the interfacial COM density(which is almost indistinguishable from the oxygen density)and hydrogen density profiles are plotted for the same externalelectric fields as in Figure 1 for the graphite and rutile surfaces.The position of the plane z = 0 corresponds to the position ofTi atoms in the interfacial Ti−O layer. Upon switching on a

strong external electric field, water molecules undergosubstantial structural changes. While the effect of a homoge-neous external field in the bulk phase is only alignment ofmolecular dipoles along the field (with no change in density),the interfacial structure also exhibits changes in local density, asa result of the inhomogeneous total electric field. Throughoutthe range of external fields applied, the changes in COMdensity profiles were mostly limited to narrowing andenhancing the density peaks. Thus, we can define a waterlayer as water molecules located between adjacent minima inCOM density, the positions of which are independent of theapplied external field. The reason for the latter requirement isthat we will subtract polarization in a layer at zero external fieldfrom that at a finite field, and therefore it is more consistent toobtain both values from the same volume. The three studiedlayers, referred to as L1, L2, and L3, are indicated in Figure 2.The layers are of molecular width except for L3 at the rutilewall. For the strongest field strengths 1.4 and 1.5 V/Å,structural changes in some layers were too dramatic leading todeviations in the position and thickness of the layer withrespect to the zero-field configuration. The results for thesefields are not analyzed below.For further determination of dielectric properties, it is

essential to calculate polarization fields in the water layersdefined above. In Figure 3, we compare polarization fieldscalculated by the charge density approach, equivalent to fullmultipole expansion, according to eq 1, and polarization fieldsgenerated by dipole moments only using eq 2 where theexpansion of the polarization density vector 3 is restricted onlyto the dipole moment density P1 (calculated according to eq11). The two profiles differ at the interface, particularly in thefirst two water layers. While the electric field is clearly notaccurately described by the point dipoles approximation on asubmolecular scale, we will compare the outcome of theelectrostatic and dielectric approaches on a scale equal to orexceeding molecular dimensions, for which the point dipoleapproximation might become reasonable.The mean polarization field Epi in layer Li, i = 1, 2, 3, or bulk

water was calculated as the average value over the intervalbetween layer boundaries, Epi = ⟨Ep(z)⟩Li, where the polar-ization field profile Ep(z) was taken as generated either by thedipole moment or by all multipole moments. Finally, to assessdielectric properties, the screening factor SF −ΔEpi/Eext, cf.,eq 8, based on the mean polarization fields within each layer,was calculated and is plotted in Figure 4 for the range ofexternal fields. A surprising result is that some of the screeningfactor values are higher than 1. We will refer to thisphenomenon as overscreening as it occurs when the magnitudeof the polarization field response exceeds the magnitude of theexternal field. An ultimate consequence is a negative value ofpermittivity as defined by eq 8. The same phenomenon wasfound previously for a dipolar liquid at a dielectric wall49 andfor SPC/E water at a diamond surface.18

The differences between the dipole and full-multipoleapproaches, manifested by the difference between therespective profiles in Figure 3, are largely alleviated by theaveraging procedure and subtraction of zero field values.Consequently, SF values are qualitatively similar for bothapproaches over the physically interesting range of smallexternal fields. Nevertheless, for quantitative purposes, the fullpolarization field arising from all multipole contributions shouldbe used.18 On the other hand, although the (atomistic) full-multipole approach gives a priori accurate profiles of the

Figure 1. Snapshots from simulations of water at rutile wall for theexternal electric fields −1, 0, and 1 V/Å. A lateral look at a block ofwater molecules above 6 × 3 replicas of the rutile (110) unit cell isshown. Titanium atoms are gray, while oxygen atoms are red. In themiddle panel, surface titanium (Ti) and bridging oxygen (BO) aredenoted. Positions of water layers L1−L3 are also indicated.

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp4128012 | J. Phys. Chem. C 2014, 118, 4818−48344822

polarization field for a given model system, the resultingdielectric response is not unambiguously decomposed intodistinct structural layer contributions (L1−L3) because thevalues of the electric field in a given layer, as calculated from thecharge density, can include contributions from atoms of watermolecules belonging to different layers. This effect is notpresent in the dipole moment approach where a contribution ofa water molecule belongs entirely to the layer where themolecule COM is located.The dielectric response is nonlinear over the entire range of

applied fields for L1 and L2. Nevertheless, the smoothtransition between screening factors for negative and positivefields enables estimation of the value in the limit of Eext = 0.The smooth transition is particularly evident for dipolarapproximation values, while values arising from the full-multipole approach show more scatter because chargecontributions are not confined to a single layer. Screeningfactor values at Eext = 0 were interpolated using third-orderpolynomials −ΔEp/Eext = c0 + c1Eext + c2Eext

2 + c3Eext3 in the

interval of field strengths within ±0.5 V/Å, as shown in Figure4. The bulk screening factor was nearly constant for fieldstrengths lower than 0.4 V/Å, and so it was fitted by a constantfunction. Fitted points were considered with weights propor-tional to the inverse square of their uncertainty, which increasesfor low field strengths due to the factor 1/Eext.Screening factor values in the limit of vanishing external field

were used to calculate permittivity values for the full-multipole(SFP, εr

P) and dipole moment (SFP1, εrP1) approaches, and are

listed in Tables 1 and 2 for the graphite and rutile surfaces.Qualitatively, permittivity varies with distance from interface in

a similar manner for both surfaces (as also observed for SFvalues), although differences between adjacent water layers aremuch more pronounced for the rutile surface. Specifically, thenegative permittivity of the overscreened contact layer (L1) isfollowed by the low positive permittivity of the second waterlayer (L2) due to its reduced screening factor. The dielectricresponse of L3 closely approaches the overscreening thresholdSF = 1, which results in large positive or negative permittivities.Note that permittivity diverges to ±∞ when the dielectricresponse exactly counterbalances the external field (SF = 1) anddecays from −∞ to less negative values as the external fieldbecomes overscreened (SF > 1). While the differences betweenL3 permittivity values for dipolar and full-multipole approachesand also between the graphite and rutile surfaces are large, inreality the calculated dielectric responses are similar, as in allcases the external field is nearly exactly canceled out by thedielectric response. Finally, the bulk permittivity values (εr = 58± 3) are in perfect agreement with literature values for theTIP4P/2005 water model55 and are independent of theapproach used to calculate the polarization field and surfacetype.Beside the results of nonequilibrium simulations, Tables 1

and 2 also contain permittivity values calculated via thefluctuation formula given in eq 10 (εr

eq), and the Kirkwoodfluctuation formula (εr

Kirkwood) given in eq 9. For simplicity, weshow only εr

eq from the dipolar approximation, assuming P = P1in eq 10. The results for εr

eq perfectly match values fromnonequilibrium simulations (εr

P1), in agreement with theoriginal studies.18,49 However, εr

eq values come from longersimulations (∼15 ns as compared to ∼6 ns for nonequilibrium

Figure 2. Center of mass (top) and hydrogen (bottom) density of interfacial water at graphite (left) and rutile (right) surfaces for the externalelectric fields −1 V/Å (• • •), 0 V/Å (−), and 1 V/Å (−−−). The vertical lines in the top figures indicate boundaries of the L1, L2, and L3 waterlayers.

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp4128012 | J. Phys. Chem. C 2014, 118, 4818−48344823

simulations). The slower convergence of the fluctuationformula is illustrated in Figure 5 where profiles of (εr

P1)−1

and (εreq)−1 from ∼6 ns runs are plotted. The plot is

complemented by profiles of (εrP)−1 to make a connection

between our results for TIP4P/2005 water at hydrophobicgraphite and hydrophilic rutile (110) surfaces, with correspond-ing profiles of SPC/E water at hydrophobic and hydrophilicdiamond surfaces shown in Figure 3 of ref 18. Thenonequilibrium profiles in Figure 5 were taken fromsimulations with the weakest external fields applied so as toapproach the limit of vanishing external field for which theequilibrium fluctuation formula was derived. Finally, theεrKirkwood permittivity values fail to reproduce the trends

obtained by the other three approaches, and thus shouldnever be used for determination of interfacial permittivity,although such results are frequently published.46 Note also thatthe Kirkwood fluctuation formula yields positive permittivity exdefinition.4.2. Effect of Solvent Molecular Structure on

Interfacial Permittivity. Here, we elucidate the molecular-level origins of the interfacial dielectric properties as manifestedby strong inhomogeneity and overscreening of the externalfield, as presented in section 4.1. We start by partitioning thepolarization field Ep of an interfacial water layer into thenumber density ρ and the molecular polarizability, the latter

described by the average z-component value of the individualwater dipole moment μz:

ε ερ μ⟨ ⟩ = − ⟨ ⟩ = − ⟨ ⟩ ⟨ ⟩E P z z z

1( )

1( ) ( )zp L

0L

0L Li i i i (13)

For simplicity, we neglect the higher multipoles of polarizationdensity P.Density variations with the external field are small relative to

the polarization field variation (Tables 3 and 4). Therefore, thescreening factor can be approximated as

ερ

μ

ερ α

≡ −Δ⟨ ⟩

≈ ⟨ ⟩Δ⟨ ⟩

≡ ⟨ ⟩

SFE

E

zz

Ez

1( )

( ) 1( )z

Lp L

ext

0L

L

ext 0L L

i

i

i

i

i i (14)

Thus, the screening factor results from two quantities: densityand the response of average molecular dipole moment alongthe field, referred to as the microscopic or molecularpolarizability62 α Δμz/Eext. The effect of density on thescreening factor is straightforward; if each water dipole changedits orientation by the same amount as in the bulk, the dielectricresponse and thus the screening factor of a given water layerwould be enhanced proportionally to its density. Molecularpolarizability accounts for the nonuniform response of the

Figure 3. Polarization field of interfacial water at graphite (left) and rutile (right) walls for the external electric fields −1, 0, and 1 V/Å. Thepolarization field generated by dipoles (−) is compared to the full polarization field generated by all multipoles (−−−). The vertical lines indicateboundaries of the L1, L2, and L3 water layers as defined in Figure 2.

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp4128012 | J. Phys. Chem. C 2014, 118, 4818−48344824

molecular dipole moment because of variable local bondingstructure among interfacial water layers, which differs from thebulk value.Rather than utilizing the field dependence of the average

molecular dipole moment, we work with the average cosine of

the angle between the water dipole moment and the surfacenormal ⟨μz(z)⟩Li/μ, where μ = 0.48 e·Å is the magnitude ofwater dipole moment (for the TIP4P/2005 model). This isconvenient because the cosine value can readily be compared toits extreme values of ±1, which correspond to the full parallel/

Figure 4. Dielectric screening in terms of the screening factor (SF = −ΔEp/Eext) generated by water in the interfacial layers (L1−L3) and in the bulkat graphite (left) and rutile (right) walls. The results based on the dipolar approximation (●) and the full multipole expression (○) are compared.Lines denote polynomial interpolations used to estimate values at Eext = 0.

Table 1. Relative Permittivity of Water in the Interfacial Layers L1−L3 at a Graphite Surface and in the Bulka

graphite surface

zmin [Å] zmax [Å] SFP SFP1 εrP εr

P1 εreq εr

Kirkwood

L1 2.14 4.75 1.23(1) 1.47(1) −4.2(2) −2.1(1) −2.2 75L2 4.75 7.81 0.94(1) 0.974(5) 17(4) 39(3) 42 44L3 7.81 11.08 0.995(7) 0.985(7) 187(552) 69(55) 71 42bulk 15 L-15 0.9828(6) 0.9832(6) 57(2) 60(2) 60 55

aResults from non-equilibrium simulations using the full-multipole approach (εrP) and the dipolar approximation (εr

P1), and from equilibriumsimulations using the fluctuation formula 10 (εr

eq) and the Kirkwood fluctuation formula 9 (εrKirkwood). Positions of the layers are given by interval

(zmin, zmax) along the z-axis. The numbers in parentheses indicate the statistical uncertainty in the last digit.

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp4128012 | J. Phys. Chem. C 2014, 118, 4818−48344825

antiparallel alignment of dipoles. The average cosine is plottedin Figure 6 (relative to the right vertical axes) together with thescreening factor (as in Figure 4 but with a uniform verticalscale). Molecular polarizability at a given Eext can be inferredfrom this figure as the slope of the secant line between theaverage cosine value at Eext and 0. A concave/linear/convexshape to the cosine dependence corresponds to a decreasing/constant/increasing molecular polarizability α, and, accordingto eq 14, also to a decreasing/constant/increasing screeningfactor. The truly linear dependence of the dipole cosine,

corresponding to a constant α, is inherently present at smallexternal fields for the bulk phase where no influence of surfaceinteraction persists. Closer to the interface, particularly in L1and L2, the local bonding network affects water dipoleorientation, producing a nonzero polarization (μz/μ ≠ 0) atzero external field and altering the dipole cosine dependencedifferently as compared to the bulk. Specifically, the alignmentof L1 dipoles in the direction of the zero-field polarization(along the positive z-axis) is hindered as compared toreorientation in the opposite direction. As a result, α decreases

Table 2. Relative Permittivity of Water in the Interfacial Layers L1−L3 at the Rutile (110) Surface and in the Bulka

rutile (110) surface

zmin [Å] zmax [Å] SFP SFP1 εrP εr

P1 εreq εr

Kirkwood

L1 2.11 2.98 1.93(5) 1.78(5) −1.08(7) −1.3(1) −1.2 33L2 2.98 4.04 0.29(5) 0.48(3) 1.4 (1) 1.9(1) 1.8 59L3 4.04 8.47 1.03(1) 1.02(1) −31(14) −69(80) −31 49bulk 15 L-15 0.9825(5) 0.9825(6) 57(2) 57(2) 58 58

aFootnotes are the same as those in Table 1.

Figure 5. Inverse permittivity of water at graphite (left) and rutile (right) surfaces calculated from the full-multipole approach (εrP)−1 (red) and from

the dipole moment approximation using nonequilibrium simulations (εrP1)−1 (green) or the equilibrium fluctuation formula (εr

eq)−1 (blue). Resultsare from simulation runs of the same length (∼6 ns).

Table 3. Dielectric Properties of Water at the Graphite Surface: Polarization Field Ep, Mass Density ρm, and Average Cosine ofthe Angle between Water Dipoles and the Surface Normal μz/μ in the Interfacial Layers L1−L3, and in the Bulk for SelectedExternal Field Strengthsa

Eext [V/Å]

−1.5 −1 −0.5 −0.1 0 0.1 0.5 1 1.5

Ep = −P1/ε0 [V/Å]L1 1.913 1.242 0.522 −0.084 −0.233 −0.378 −0.902 −1.465 bL2 1.476 0.983 0.497 0.108 0.011 −0.087 −0.481 −0.985 bL3 1.473 0.984 0.494 0.102 0.005 −0.094 −0.486 −0.977 −1.469bulk 0.000 −0.098 −0.491 −0.981 −1.467

ρm [g/cm3]L1 1.24 1.16 1.08 1.09 1.10 1.11 1.15 1.20 bL2 1.00 0.98 1.00 1.02 1.02 1.02 1.02 1.01 bL3 1.00 0.99 1.00 1.00 1.00 1.00 1.00 0.99 1.00bulk 1.00 1.00 1.00 0.99 1.00

μz/μL1 −0.530 −0.368 −0.166 0.027 0.073 0.117 0.270 0.419 bL2 −0.506 −0.343 −0.171 −0.037 −0.004 0.029 0.162 0.336 bL3 −0.507 −0.342 −0.171 −0.035 −0.002 0.032 0.168 0.339 0.505bulk 0.000 0.034 0.169 0.340 0.506

aThe screening factor SF, cf., eq 8 and Figures 4 and 6, is calculated by subtraction of the polarization field in a column for a given Eext and in thecentral column for Eext = 0. bData were not analyzed due to significant modification of the density profiles and layer boundaries.

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp4128012 | J. Phys. Chem. C 2014, 118, 4818−48344826

markedly for positive external fields and, equivalently, SFdecreases more steeply on the positive branch of external fields.In the case of the rutile surface, the decrease of SF in L1 forpositive fields is enhanced due to saturation of the dipole cosinewhen close to its maximum value +1. This decrease is balancedby a comparable increase in L2, where the external field ispoorly screened by L1, whereas for negative and low positiveexternal fields α is low, resulting in low SF. The same effect isalso noticeable at the graphite interface, although it is much lesspronounced (see the detailed dependence of the screeningfactor in Figure 4). The mild interaction of the graphite surfacewith interfacial water layers is reflected by the roughly lineardependence of the average cosine with substantial deviationsonly in L1. In addition, the zero field cosine is small in allinterfacial water layers, in agreement with previous findings thatwater molecules orient primarily parallel to the graphitesurface.63 On the other hand, the influence of rutile−waterinteractions persists markedly in all interfacial water layers asmanifested by the series of oppositely oriented dipolemoments: μz/μ = 0.76, −0.52, and 0.05 in L1, L2, and L3,respectively, for Eext = 0.In the previous paragraph, the dependence of the average

cosine of the interfacial water layers was discussed for the entirerange of applied fields. However, for the determination ofpermittivity at experimentally relevant conditions, only theslope of the dependence around Eext = 0 is important. In Figure7, a comparison of the slopes, that is, molecular polarizabilitiesαLi, at Eext = 0 in the three water layers as well as the bulk isgiven. For the graphite interface, only the molecular polar-izability of L1 differs markedly from the bulk value, while forthe rutile interface, L2 and L3 exhibit deviating molecularpolarizability (slightly increased in L3, and significantly reducedin L2). The ratios of layer susceptibilities to the bulk are givenin Table 5 . Given that the densi ty ra t ios are⟨ρ⟩L1:⟨ρ⟩L2:⟨ρ⟩L3:⟨ρ⟩bulk = 1.10:1.02:1.00:1.00 for graphite,and 1.78:1.46:0.94:1.00 for rutile interfaces, the consequencesof eq 14 for screening factors and permittivities are as follows.For the graphite interface, the screening factor is a combinedeffect of density and molecular polarizability. For the rutileinterface, the overscreening of L1 is a direct consequence of theincreased density, while the poor screening of L2 and the slightoverscreening of L3 arise solely from a reduced and slightly

enhanced molecular polarizability, respectively. Although thedielectric profile at the two interfaces is qualitatively the same,that is, an overscreening response (SF > 1, negativepermittivity) for L1, a reduced response (SF < SFbulk, lowpositive permittivity) for L2, and a slightly enhanced responsefor L3 (SF ≥ SFbulk, singular permittivity), we can now see thatthe mechanisms underlying the dielectric response are different.So far, the dielectric response has been described rather

phenomenologically. In the following, we elucidate howmolecular polarizability arises from the local bonding network.Molecular polarizability α reflects the ease of alignment of amolecular dipole along an applied field. On a molecular scale,polarizability depends on the local mean force acting on amolecule toward an applied field, which arises from thedistortion of an equilibrium bonding network. In Tables 6 and7, the number of bonds between the studied structural layersand their energy are given. Bonds are defined to form betweenoppositely charged sites of different molecules, which includeswater−water hydrogen bonds and bonds with surface groups inthe case of rutile: hydrogen bonds between water hydrogen andBO and between water oxygen and Ti. Bonds were identifiedon the basis of a geometric criterion; that is, a bond exists whenthe interaction sites are closer than a maximum bond lengthrmax, defined as the position of the first minimum in thecorresponding site−site radial distribution function. The bondcharacteristics are surveyed in Table 8, and bonds are visualizedin Figure 8. The energy of bonds was sampled directly in thesimulation, together with calculation of forces betweeninteracting sites.For the graphite surface (Table 6), the energy (strength) of

hydrogen bonds between various structural layers is similar.Hence, the resistance of a water molecule to reorientation isrelated to the number of bonds the molecule forms(coordination number). Comparison of the number of bondsper molecule in the layers L1−L3 with correspondingmolecular polarizability from Table 5 (and Figure 7) confirmsthat the enhanced molecular polarizability of L1 is caused bythe reduced number of bonds, which eases the alignment ofmolecular dipoles in an applied field relative to the bulk. Thecoordination number in subsequent layers is similar to the bulkvalue with a slightly increased value in L2, again in line with thereduced molecular polarizability.

Table 4. Dielectric Properties of Water at the Rutile (110) Surfacea

Eext [V/Å]

−1.5 −1 −0.5 −0.1 0 0.1 0.5 1 1.5

Ep = −P1/ε0 [V/Å]L1 −0.871 −1.844 −2.907 −3.752 −3.941 −4.107 −4.572 −4.791 −4.892L2 2.891 2.656 2.399 2.245 2.199 2.145 1.839 1.091 0.050L3 1.359 0.867 0.373 −0.036 −0.141 −0.246 −0.632 −1.076 bbulk 0.000 −0.098 −0.491 −0.981 −1.467

ρm [g/cm3]L1 1.78 1.78 1.78 1.78 1.78 1.78 1.78 1.78 1.78L2 1.45 1.45 1.45 1.46 1.46 1.46 1.46 1.44 1.37L3 0.95 0.94 0.94 0.94 0.94 0.95 0.95 0.97 bbulk 1.00 1.00 1.00 1.00 1.00

μz/μL1 0.169 0.357 0.562 0.725 0.762 0.794 0.884 0.926 0.945L2 −0.685 −0.630 −0.569 −0.531 −0.519 −0.506 −0.433 −0.261 −0.013L3 −0.495 −0.319 −0.137 0.013 0.053 0.089 0.228 0.383 bbulk 0.000 0.034 0.169 0.340 0.506

aFootnotes are the same as those in Table 3.

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp4128012 | J. Phys. Chem. C 2014, 118, 4818−48344827

In the case of the rutile surface (Table 7), the bondingnetwork is more complex as strong bonds with the surface exist.Hence, the number of bonds has to be analyzed together withtheir strength. Although the number of bonds per molecule inL1 is lower than in bulk, their strength limits the alignment of aL1 water molecule, so that the resulting molecular polarizabilityis the same as for the bulk (this similarity is coincidental). Thedense bonding network around L2 water molecules (our simpledefinition of a bond allows “overbonding”, i.e., number ofbonds per water molecule >4) is supported by high bondingstrength, which together are responsible for the low α. For L3,the increased α seems to be a prevailing effect of a reduced

bond strength rather than a slightly increased coordinationnumber.

4.3. Electrostatic Profile at Charged Surfaces andCapacitance of the Stern Layer. In this section, we will linkMD simulation results to the surface complexation modeling ofmacroscopic experimental data. To study EDL structure, asimulation of electrolyte contacting a charged rutile surface wascarried out, and the electrostatic profile was analyzed. Next, thecapacitance of the Stern layer, which is one of the key surfacecomplexation variables, will be determined using thepermittivity profile of water determined in section 4.1.Simulation of the negatively charged rutile (110) surface was

carried out by adding hydroxyl groups to terminal Ti atoms and

Figure 6. Screening factor (SF) (●, left vertical axes) and average cosine of the angle between water molecule dipole moment and the surface normal(○, right vertical axes) in the interfacial water layers (L1−L3) and in bulk water for graphite (left) and rutile (right) surfaces.

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp4128012 | J. Phys. Chem. C 2014, 118, 4818−48344828

redistributing their charge over the surface as described in ref10. The total surface charge density was ∼−0.1 C/m2 and wasneutralized by excess Na+ ions of the background NaClelectrolyte (0.35 M). The electrostatic potential profile andcumulative charge density are given in Figure 9. The

electrostatic potential ψ was calculated by the followingequation:

∫ψε

ρ= − ′ ′ − ′−∞

z z z z z( )1

d ( )( )z

q0 (15)

Hence, both quantities stem from the charge density ρq, whichcan be decomposed into contributions from surface atoms,ions, and water. In Figure 9, the full profiles (solid lines), theprofiles generated by the ions plus the surface charge (dashedlines), and those generated by pure water at the neutral rutilesurface (dotted lines) are shown. Interestingly, the rapidoscillations of the full potential profile at the charged surface arenicely matched by the profile for pure water at the neutralsurface. Comparison of the two profiles derived fromelectrolyte solution reveals that the slowly decaying ions +surface potential is largely compensated by the watercontribution, which causes the full potential to be muchmore short ranged. This is further evidence, in addition tostudies of interfacial diffusivity15 and viscosity,16 that theeffective range of surface−solution or solute−solvent inter-actions in aqueous solutions is only about 15 Å, in contrast totheories based on implicit solvent models that predict aneffective surface interaction range (such as the position of the“shear plane”) of several nanometers, depending on the ionicstrength. The dissimilarity between full electrostatic profile andthat generated by ions + surface charge shows that waterdielectric properties cannot be described accurately by simplemodels, which neglect its molecular structure. The cumulativecharge density reveals that the surface charge is overscreened bythe ionic charge beyond the adsorption plane of cations (∼3.5Å). This phenomenon has been observed previously for bothhydrophilic and hydrophobic surfaces.64 Throughout thisArticle, the term “overscreening” refers to the total electricfield response having a different sign than the external field. In abroader sense, which covers the case of solutions containingions, the external field can be considered as generated by chargedensity excluding the solvent contribution, that is,

∫ ρ= ′ ′ =ε ε−∞ +E z zd ( )

z Dext

1q,ions surface0 0

. From this, “overscreen-

ing” can be defined as the total electric field response ΔEhaving a sign opposite to the displacement field, D/ε0. Thedashed line in Figure 9 (bottom) is equal to the displacementfield D, and the difference between the full (solid) and neutral

surface (dotted) profiles is equal to ∫ ρΔ ′ ′ =ε−∞

Δz zd ( )z

qE

0.

Figure 7. Response of a water molecule’s dipole orientation to the applied field in the interfacial water layers L1 (red), L2 (green), L3 (blue) and inbulk (black) for graphite (left) and rutile (right) interfaces. Simulation data, taken from Figure 6, were fitted by linear function a*Eext in the intervalEext ∈ [−0.1, 0.1] V/Å. The slope a = α/μ yields molecular polarizability α.

Table 5. Density, Molecular Polarizability, and ScreeningFactor (SFP1) of the Interfacial Water Layers L1−L3Normalized by the Bulk Valuesa

graphite rutile

ρLi/ρbulk αLi/αbulk SFLi/SFbulk ρLi/ρbulk αLi/αbulk SFLi/SFbulk

L1 1.10 1.34 1.495 1.78 1.01 1.812L2 1.02 0.98 0.991 1.46 0.36 0.489L3 1.00 1.00 1.002 0.94 1.13 1.038

aThe normalized values of density and molecular polarizabilityrepresent their effect on the resulting screening factor, cf., eq 14.

Table 6. Number and Energy of Hydrogen Bonds betweenStructural Layers at the Graphite Surfacea

no. of bonds

L1 L2 L3 rest bulkbonds/molecule

L1 633.3 240.0 3.30L2 591.7 240.7 3.74L3 634.6 242.6 3.71rest 4069.5 242.9 3.71bulk 3177.3 3.71

bond energy [kJ/mol]

L1 L2 L3 rest bulk averageL1 −19.3 −18.7 −19.1L2 −18.9 −18.8 −18.8L3 −19.0 −18.9 −18.9rest −18.9 −18.9 −18.9bulk −18.9 −18.9

aThe number of bonds per molecule, a measure of its resistance toreorientation, explains molecular polarizability α given in Table 5: thehigher is the number of bonds per molecule, the lower is the molecularpolarizability. Structural layers are defined as in Table 1 with “rest”being a region residing between L3 and bulk. The absolute number ofbonds refers to the number of water molecules in the unit simulationcell (265, 287, 301, 1227, and 921 molecules in L1, L2, L3, rest, andbulk, respectively).

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp4128012 | J. Phys. Chem. C 2014, 118, 4818−48344829

Therefore, it can be seen that the region of overscreening (inthe above-defined sense) does not coincide with the region ofoverscreening of the surface charge by the ionic charge.Therefore, the two different overscreening phenomena are notdirectly related.The electrostatic profile at a charged surface gives rise to

capacitance. It is tempting to calculate capacitance directly fromFigure 9 as the quotient of surface charge density σ0 oversurface potential Ψ0. Note, however, that a nontrivial surface

potential exists even for a neutral surface due to the interfacial

structure of water (dotted line, Figure 9 (top)). Following the

discussion of the derivation of interfacial permittivity, the

capacitance value that can be linked to experimental data is

Table 7. Number and Energy of Bonds between Structural Layers at the Rutile (110) Surfacea

no. of bonds

surface L1 L2 L3 rest bulk bonds/molecule

L1 173.0 123.3 117.2 82.9 3.45L2 142.9 178.9 166.3 4.17L3 941.6 244.7 3.73rest 4735.7 242.6 3.71bulk 3173.2 3.71

bond energy [kJ/mol]

surface L1 L2 L3 rest bulk averageL1 −69.1 −7.5 −18.2 −16.8 −33.0L2 −41.5 −14.1 −18.3 −22.5L3 −18.8 −18.8 −18.7rest −18.9 −18.9 −18.9bulk −18.9 −18.9

aThe number of bonds per molecule and their energy govern resistance of a molecule to reorientation, and can be used to account for the molecularpolarizability α given in Table 5. Structural layers are defined as in Table 2 with “rest” being a region residing between L3 and bulk. The absolutenumber of bonds refers to the number of water molecules in the unit simulation cell (144, 145, 385, 1406, and 920 molecules in L1, L2, L3, rest, andbulk, respectively).

Table 8. Bonds Are Defined To Form between OppositelyCharged Sites Distanced Less than rmax

a

bond type rmax [Å]

HW−OW 2.43HW−BO 2.43OW−Ti 3.23

aHydrogen bonds form between the positive (HW) and negative(OW) sites on different water molecules; bonds with the rutile (110)surface involve bridging oxygen (BO) or terminal titanium (Ti).

Figure 8. Bonding network at the rutile (110)−water interface.Hydrogen bonds between water molecules and those with bridgingoxygens (BO) of the surface are shown in blue, while bonds withsurface titanium atoms (Ti) are magenta. The figure covers the sameregion as in Figure 1.

Figure 9. Electrostatic potential (top) and cumulative charge density(bottom) of 0.35 M NaCl electrolyte in contact with a negativelycharged rutile (110) surface of surface charge density ≈ −0.1 C/m2.The full profiles (−) are compared to those due to the distribution ofions (−−−), that is, lacking the contribution of water, and todistributions for pure water at a neutral (for Eext = 0) rutile surface(· · ·).

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp4128012 | J. Phys. Chem. C 2014, 118, 4818−48344830

σψ

σψ

=ΔΔ

=c cordd

0

0

0

0 (16)

Calculation of capacitance according to eq 16 directly fromsimulation would require having a consistent molecular modelfor a variable charged surface. Here, we calculate capacitancefrom the permittivity of water, as determined in section 4.1,exploiting the following chain of equations:

∫ ∫ ∫σψ

σ ε σε

εε

=ΔΔ

=Δ

Δ≈

ΔΔ

≈− −cE z D z zd d d

0

0

0 0 0

r1

0

r1

(17)

where we assume that presence of ions does not affect solventpermittivity and that changing the surface charge (at constantnumber of ions) does not affect ionic distribution, so thatΔD(z) = Δσ0. Neglecting the impact of ions on permittivityimplies the capacitance is independent of electrolyte type andionic strength. Such an approximation is also widely used insurface complexation modeling where various EDL models arefit to experimental surface titration data for various electrolytesand ionic strengths to arrive at a unique capacitance value foreach Stern layer included.29,35 The other approximation, ΔD(z)= Δσ0, is reasonable when the change in cumulative ioniccharge distribution is small as compared to added surface

charge, ∫ ρ σ′Δ ′ < Δ−∞

z zd ( )z

q,ions 0. The change in Δρq is

sometimes substantial, for example, Na+ moving from abidentate to a tetradentate binding mode with increasingnegative surface charge.64 Nevertheless, these changes might becounterbalanced when integrated over the Stern layer, whichwe are interested in. In addition to the discussed assumptions,the local relation between the electric and displacement fields,D(z) = εr(z)ε0E(z), was used, which is justified provided thedisplacement field varies much slower than the electric field, asevident from Figure 9 (bottom).The Stern layer capacitance is defined as

∫σ

ψ ψεε

=Δ

Δ −≈

−c

z( ) dz

zS0

0 S

0

r1

0

S

(18)

where ΨS is the electrostatic potential at the Stern plane locatedat distance zS from the surface. In surface complexationmodeling, cS plays the key role as a general characteristic of thesolid−solution interface. Ridley et al.65 fit experimental titrationdata for rutile in several 1−1 electrolytes at ambient conditionswith the basic Stern model, and the resulting Stern layercapacitance was about 0.6 F/m2. Here, we calculated thecapacitance according to eq 18 using the permittivity profile εr

P

shown in Figure 5 with the inverse permittivity integratedbetween the surface charge and Stern planes. The Stern planeposition at zS = 3.5 Å was inferred from Figure 9 assuming thatit coincides with the peak position of Na+ counterions.However, the position of the surface charge plane z0 is moreambiguous and does not necessarily coincide with z = 0,defined as the position of interfacial Ti atoms (see Figure 1)because bridging oxygen atoms (BO) and terminal hydroxylgroups (TOH) also carry charge. These BO and TOH groupsare located 1.14 and 1.87 Å above the Ti atom surface plane,

respectively. The running integral ∫ ε ′ ′− z z( ) dz

0 r1 is plotted in

Figure 10, and the value of the integral between boundaries a(z0) and b (zS) is given as a difference between the values onthe curve at b and a. The effect of varying the integrationboundaries on the capacitance is shown in the inset where the

upper boundary is fixed at zS = 3.5 Å, while the lower boundaryvaries between z0 = 0 (position of Ti) and z0 = 1.87 Å (positionof TOH). The calculated capacitance strongly depends on thechosen position of the surface charge plane, and it increases as1/(z − z0) when approaching z0 = 1.83 Å (just before theposition of TOH). At that position, the integral in eq 18vanishes, and the calculated capacitance is negative at greaterdistances because of the dominant influence of the L1 waterlayer, which has negative permittivity. However, negativecapacitances have never been observed experimentally, whichsuggests that the negative permittivity (if it exists at all; notethat ionization effects and electronic polarizability are neglectedin our molecular models) is compensated by the positivepermittivity of the depletion layer (between the surface andwater) where εr

−1 = 1. The value of cS = 0.6 F/m2, determinedby surface complexation modeling, corresponds to z0 = 1.7 Å,that is, between the BO and TOH surface groups. This isreasonable because both groups carry charge and participate inbinding ionic species. Moreover, most solid surfaces are notperfectly planar and are populated by charged surface groups atdifferent heights, so locating the surface charge plane (z0) willoften be ambiguous.

4.4. Significance of Solvent Molecular Structure. Ourresults of the distance-dependent permittivity at a solid−liquidinterface are in contrast with the implicit solvent model picture,where the relative permittivity of interfacial water typicallydecreases monotonically toward the surface from bulk values tolow positive values at the interface.26,27,29,31,37,38 A criticalshortcoming of implicit solvent models is that they account forinterfacial dielectric properties with the same relations used forhomogeneous systems. Consequently, the permittivity ofinterfacial water is lower than in bulk due to saturation of thewater dipoles according to a relation between field strength andpermittivity of a homogeneous continuum, such as the Boothrelation:31,66,67

ε ε= + − −

= + βμ

⎡⎣⎢

⎤⎦⎥E n n

bEbE

bE

b n

( ) ( (0) )3

coth( )1

,

736

( 2)

r2

r2

2(19)

Figure 10. Running integral of inverse permittivity εrP taken from

Figure 5. The positions of surface groups are indicated by verticallines: Ti = interfacial titanium atoms, BO = bridging oxygen, TOH =oxygen of hydroxyl group of a negatively charged rutile surface. Theeffect of position of the surface charge plane z0 on the Stern layercapacitance cS is shown in the inset. The dashed line indicates the valueobtained by surface complexation modeling of experimental rutiletitration data as presented in ref 65.

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp4128012 | J. Phys. Chem. C 2014, 118, 4818−48344831

where n is the refractive index. In contrast, our results revealthat the dielectric responses of interfacial and bulk watermolecules to an external field are different (see Figures 4 and6).This discrepancy arises because implicit solvent models do

not consider spatial variations in solvent structure. For example,eq 19 only accounts for a permittivity decrease from saturationof the dielectric response when dipoles are aligned with thefield. However, our simulation results demonstrate thatvariations in density and molecular polarizability due to thepresence of surfaces also have a significant effect on thedielectric response. For the strongly hydrophilic rutile surface,these surface-induced variations increase (in L1) or decrease(in L2) the dielectric response by a factor of 2 relative to thebulk response. A similar effect is evident also for the weaklyinteracting graphite surface where the locally decreasedmolecular polarizability (and increased density), resultingfrom the lack of bonding partners, leads to a dielectric response(of L1) enhanced by a factor 1.5 relative to bulk water. In bothcases, the effect of interfacial water structure leads tooverscreening of the external field by the dielectric response,a phenomena that does not occur in bulk water at any fieldstrength and therefore cannot be described by bulk relations,such as eq 19.This has an important implication for theoretical models of

interfaces: if density and molecular polarizability variations ofthe solvent are significant enough to cause overscreening of theexternal field, then the solvent contribution to the electrostaticpotential profile and other interfacial dielectric properties canno longer be considered as secondary to ionic contributions. Asshown in Figure 9, the structure of water molecules activelyforms and dominates the electrostatic profile rather thandiminishing the electric field radiated by the rutile (110) surfaceand ions.

5. CONCLUSIONS

We applied nonequilibrium molecular simulations to study thedielectric response of interfacial water perturbed by externalelectric fields normal to the interface. Analysis of water layers atweakly interacting (hydrophobic) graphite and stronglyinteracting (hydrophilic) rutile (110) surfaces revealed theeffect of solvent molecular structure induced by the surface ondielectric properties, which is neglected in implicit solventmodels. The key findings can be summarized as follows.

(a) The dependence of the dielectric response on theexternal field in the interfacial region differs entirely fromthe response in bulk (homogeneous) water, leading tostrong deviations in permittivity. A particularly strongand weak dielectric response was found in the contactand second hydration layer, respectively.

(b) The dielectric response was found to arise from localdensity and molecular polarizability, that is, the responsein orientation of an individual molecule. The interplaybetween both factors means that that preferentialalignment of dipoles does not necessarily lead to areduced dielectric response as commonly assumed on thebasis of bulk behavior. On the contrary, an enhanced oreven an overscreening response, leading to negativepermittivity, can occur in a local part of the system,particularly at the interface, where multiple interactionsinterplay. The permittivity of a macroscopic continuoussystem must be larger than unity in accordance with

stability laws; however, negative permittivity in a finitepart of an inhomogeneous system does not violate theselaws. Moreover, such localized phenomena cannot beaccounted for by the bulk-like relations used in implicitsolvent models.

(c) The molecular level origin of the dielectric responseresides in the solvent structure determined by the localdensity and H-bonding network. Molecular polarizabilityshows a negative correlation with number of bonds permolecule and bond strength.

(d) The interfacial dielectric properties are qualitativelysimilar for both the graphite (hydrophobic) and therutile (110) (hydrophilic) surfaces, although the under-lying mechanisms are a bit different. The overscreeningresponse of the contact layer is an effect of augmenteddensity at the rutile surface, while it results more fromincreased molecular polarizability due to the lack of H-bonding partners at the graphite surface. The limitedmolecular polarizability of the densely bonded secondwater layer is responsible for its reduced response andlow permittivity at both surfaces, but especially for rutile.

(e) The equilibrium fluctuation formulas of Ballenegger etal.49 and Bonthuis et al.18 yield excellent agreement withthe nonequilibrium approach using an external field. Onthe other hand, the Kirkwood fluctuation formula, whichwas derived for homogeneous isotropic fluids, should notbe used to calculate interfacial permittivities.

(f) The results obtained for neutral surfaces in contact withpure water were compared to a simulation of a negativelycharged rutile surface interacting with 0.35 M NaCl. Thecorresponding electrostatic potential profiles were verysimilar, which supports the “water centric” as opposed tothe “ionic centric” concept of electrical double layers, asthe dominant contribution comes from the chargedensity distribution of water.

(g) The calculated permittivity profile at the rutile (110)surface yields a Stern layer capacitance value compatiblewith that obtained from surface complexation modelingof experimental titration data provided the surface chargeplane is set to reside between the two charged surfacegroups, which reside at different heights. The ambiguityin setting the location of the surface charge plane foratomically rough surfaces is the main limitation inpredicting the capacitance of interfacial layers fromcomputer simulations.

Future studies toward a better understanding of interfacialdielectric properties should address the impact of ions onpermittivity and utilize polarizable simulation models.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: parez@icpf.cas.cz.NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThis research was supported by the Czech Science Foundation(project nos. 13-08651S and 13-09914S). Access to theMetaCentrum computing facilities, provided under theprogramme “Projects of Large Infrastructure for Research,Development, and Innovations” LM2010005 funded by theMinistry of Education, Youth, and Sports of the Czech

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp4128012 | J. Phys. Chem. C 2014, 118, 4818−48344832

Republic, is acknowledged. M.M. was supported by the U.S.Department of Energy, Office of Science, Basic EnergySciences, Chemical Sciences, Geosciences, and BiosciencesDivision, to help interpret the MD results and relate them tomacroscopic data.

■ REFERENCES(1) Kavathekar, R. S.; Dev, P.; English, N. J.; MacElroy, J. M. D.Molecular Dynamics Study of Water in Contact with the TiO2 Rutile-110, 100, 101, 001 and Anatase-101, 001 Surface. Mol. Phys. 2011, 109,1649−1656.(2) Koparde, V. N.; Cummings, P. T. Molecular Dynamics Study ofWater Adsorption on TiO2 Nanoparticles. J Phys. Chem. C 2007, 111,6920−6926.(3) Wang, J.; Kalinichev, A. G.; Kirkpatrick, R. J. Effects of SubstrateStructure and Composition on the Structure, Dynamics, andEnergetics of Water at Mineral Surfaces: A Molecular DynamicsModeling Study. Geochim. Cosmochim. Acta 2006, 70, 562−582.(4) Wolthers, M.; Di Tommaso, D.; Du, Z.; de Leeuw, N. H. CalciteSurface Structure and Reactivity: Molecular Dynamics Simulations andMacroscopic Surface Modelling of the Calcite-Water Interface. Phys.Chem. Chem. Phys. 2012, 14, 15145−15157.(5) Philpott, M. R.; Goliney, I. Y.; Lin, T. T. Molecular DynamicsSimulation of Water in a Contact with an Iron Pyrite FeS2 Surface. J.Chem. Phys. 2004, 120, 1943−1950.(6) Tournassat, C.; Chapron, Y.; Leroy, P.; Bizi, M.; Boulahya, F.Comparison of Molecular Dynamics Simulations with Triple Layerand Modified Gouy-Chapman Models in a 0.1M NaCl-Montmor-illonite System. J. Colloid Interface Sci. 2009, 339, 533−541.(7) Bourg, I. C.; Sposito, G. Molecular Dynamics Simulations of theElectrical Double Layer on Smectite Surfaces Contacting Concen-trated Mixed Electrolyte (NaCl-CaCl2) Solutions. J. Colloid InterfaceSci. 2011, 360, 701−715.(8) Rosenqvist, J.; Machesky, M. L.; Vlcek, L.; Cummings, P. T.;Wesolowski, D. J. Charging Properties of Cassiterite (α-SnO2)Surfaces in NaCl and RbCl Ionic Media. Langmuir 2009, 25,10852−10862.(9) Katz, L. E.; Criscenti, L. J.; Chen, C.-c.; Larentzos, J. P.;Liljestrand, H. M. Temperature Effects on Alkaline Earth Metal IonsAdsorption on Gibbsite: Approaches from Macroscopic SorptionExperiments and Molecular Dynamics Simulations. J. Colloid InterfaceSci. 2013, 399, 68−76.(10) Predota, M.; Bandura, A. V.; Cummings, P. T.; Kubicki, J. D.;Wesolowski, D. J.; Chialvo, A. A.; Machesky, M. L. Electric DoubleLayer at the Rutile (110) Surface. 1. Structure of Surfaces andInterfacial Water from Molecular Dynamics by Use of ab InitioPotentials. J. Phys. Chem. B 2004, 108, 12049−12060.(11) Predota, M.; Zhang, Z.; Fenter, P.; Wesolowski, D. J.;Cummings, P. T. Electric Double Layer at the Rutile (110) Surface.2. Adsorption of Ions from Molecular Dynamics and X-rayExperiments. J. Phys. Chem. B 2004, 108, 12061−12072.(12) Predota, M.; Vlcek, L. Comment on Parts 1 and 2 of the Series“Electric Double Layer at the Rutile (110) Surface. J. Phys. Chem. B2007, 111, 1245−1247.(13) Zhang, Z.; Fenter, P.; Cheng, L.; Sturchio, N. C.; Bedzyk, M. J.;Machesky, M. L.; Wesolowski, D. J. Model-Independent X-ray Imagingof Adsorbed Cations at the Crystal-Water Interface. Surf. Sci. 2004,554, L95−L100.(14) Zhang, Z.; Fenter, P.; Cheng, L.; Sturchio, N. C.; Bedzyk, M. J.;Predota, M.; Bandura, A.; Kubicki, J.; Lvov, S. N.; Cummings, P. T.;et al. Ion Adsorption at the Rutile-Water Interface: Linking Molecularand Macroscopic Properties. Langmuir 2004, 20, 4954−4969.(15) Predota, M.; Cummings, P. T.; Wesolowski, D. J. ElectricDouble Layer at the Rutile (110) Surface. 3. Inhomogeneous Viscosityand Diffusivity Measurement by Computer Simulations. J. Phys. Chem.C 2007, 111, 3071−3079.

(16) Parez, S.; Predota, M. Determination of Distance-dependentViscosity of Mixtures in Parallel Slabs using Non-equilibriumMolecular Dynamics. Phys. Chem. Chem. Phys. 2012, 14, 3640−3650.(17) Machesky, M. L.; Predota, M.; Wesolowski, D. J.; Vlcek, L.;Cummings, P. T.; Rosenqvist, J.; Ridley, M. K.; Kubicki, J. D.;Bandura, A. V.; Kumar, N.; et al. Surface Protonation at the Rutile(110) Interface: Explicit Incorporation of Solvation Structure withinthe Refined MUSIC Model Framework. Langmuir 2008, 24, 12331−12339.(18) Bonthuis, D. J.; Gekle, S.; Netz, R. R. Profile of the StaticPermittivity Tensor of Water at Interfaces: Consequences forCapacitance, Hydration Interaction and Ion Adsorption. Langmuir2012, 28, 7679−7694.(19) McCafferty, E.; Zettlemoyer, A. C. Adsorption of Water Vapouron α-Fe2O3. Discuss. Faraday Soc. 1971, 52, 239−254.(20) Wachter, W.; Buchner, R.; Hefter, G. Hydration ofTetraphenylphosphonium and Tetraphenylborate Ions by DielectricRelaxation Spectroscopy. J. Phys. Chem. B 2006, 110, 5147−5154.(21) Heugen, U.; Schwaab, G.; Brundermann, E.; Heyden, M.; Yu,X.; Leitner, D. M.; Havenith, M. Solute-induced Retardation of WaterDynamics Probed Directly by Terahertz Spectroscopy. Proc. Natl.Acad. Sci. U.S.A. 2006, 103, 12301−12306.(22) Tielrooij, K. J.; Paparo, D.; Piatkowski, L.; Bakker, H. J.; Bonn,M. Dielectric relaxation Dynamics of Water in Model MembranesProbed by Terahertz Spectroscopy. Biophys. J. 2009, 97, 2484−2492.(23) Teschke, O.; Ceotto, G.; de Souza, E. F. Dielectric ExchangeForce: A Convenient Technique for Measuring the Interfacial WaterRelative Permittivity. Phys. Chem. Chem. Phys. 2001, 3, 3761−3768.(24) Bernik, D. L.; Negri, R. M. Local Polarity at the Polar HeadLevel of Lipid Vesicles Using Dansyl Fluorescent Probes. J. ColloidInterface Sci. 1998, 203, 97−105.(25) Bernik, D. L.; Negri, R. M. Critical Revision of ApparentDielectric Constants Calculations in Lipid−Water Interfaces. J. ColloidInterface Sci. 2000, 226, 364−366.(26) Hiemstra, T.; van Riemsdijk, W. H. Encyclopedia of Colloid andInterface Science; Marcel Dekker: New York, 2002.(27) Hiemstra, T. Surface Complexation at Mineral Interfaces:Multisite and Charge Distribution approach. Ph.D. Thesis, Wagenin-gen University, 2010.(28) Brown, G. E., Jr.; Henrich, V. E.; Casey, W. H.; Clark, D. L.;Eggleston, C.; Felmy, A.; Goodman, D. W.; Gratzel, M.; Maciel, G.;McCarthy, M. I.; et al. Metal Oxide Surfaces and Their Interactionswith Aqueous Solutions and Microbial Organisms. Chem. Rev. 1999,99, 77−174.(29) Hiemstra, T.; Van Riemsdijk, W. H. On the RelationshipBetween Charge Distribution, Surface Hydration, and the Structure ofthe Interface of Metal Hydroxides. J. Colloid Interface Sci. 2006, 301,1−18.(30) Stern, O. The Theory of the Electrolytic Double-Layer. Z.Elektrochem. Angew. Phys. Chem. 1924, 30, 508−516.(31) Wang, H.; Pilon, L. Accurate Simulations of Electric DoubleLayer Capacitance of Ultramicroelectrodes. J. Phys. Chem. C 2011, 115,16711−16719.(32) Masliyah, J. H.; Bhattacharjee, S. Electrokinetic and ColloidTransport Phenomena; John Wiley and Sons: Hoboken, NJ, 2006.(33) Lyklema, J. Fundamentals of Interface and Colloid Science, Vol. IISolid-Liquid Interfaces; Academic Press: New York, 1995.(34) Rahnemaie, R.; Hiemstra, T.; Van Riemsdijk, W. H. A NewSurface Structural Approach to Ion Adsorption: Tracing the Locationof Electrolyte Ions. J. Colloid Interface Sci. 2006, 293, 312−321.(35) Sverjensky, D. A. Interpretation and Prediction of Triple-LayerModel Capacitances and the Structure of the Oxide-Electrolyte-WaterInterface. Geochim. Cosmochim. Acta 2001, 65, 3643−3655.(36) Sverjensky, D. A. Prediction of Surface Charge on Oxides in SaltSolutions: Revisions for 1:1 (M+L-) Electrolytes. Geochim. Cosmochim.Acta 2005, 69, 225−257.(37) Frydel, D.; Oettel, M. Charged Particles at Fluid Interfaces as aProbe into Structural Details of a Double Layer. Phys. Chem. Chem.Phys. 2011, 13, 4109−4118.

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp4128012 | J. Phys. Chem. C 2014, 118, 4818−48344833

(38) Wang, H.; Varghese, J.; Pilon, L. Simulation of Electric DoubleLayer Capacitors with Mesoporous Electrodes: Effects of Morphologyand Electrolyte Permittivity. Electrochim. Acta 2011, 56, 6189−6197.(39) Vorotyntsev, M. A.; Mityushev, P. V. Electric Double LayerStructure in a Surface-Inactive Electrolyte Solution: Effect of the SternLayer and Spatial Correlations of Solvent Polarization. Electrochim.Acta 1991, 36, 401−409.(40) Finken, R.; Ballenegger, V.; Hansen, J.-P. Onsager Model for aVariable Dielectric Permittivity near an Interface. Mol. Phys. 2003, 101,2559−2568.(41) Ghoufi, A.; Szymczyk, A.; Renou, R.; Ding, M. Calculation ofLocal Dielectric Permittivity of Confined Liquids from Spatial DipolarCorrelations. Europhys. Lett. 2012, 99, 37008.(42) Lorenz, C. D.; Crozier, P. S.; Anderson, J. A.; Travesset, A.Molecular Dynamics of Ionic Transport and Electrokinetic Effects inRealistic Silica Channels. J. Phys. Chem. C 2008, 112, 10222−10232.(43) Pandit, S. A.; Bostick, D.; Berkowitz, M. L. An Algorithm toDescribe Molecular Scale Rugged Surfaces and Its Application to theStudy of a Water/Lipid Bilayer Interface. J. Chem. Phys. 2003, 119,2199−2205.(44) Lin, J.-H.; Baker, N. A.; McCammon, J. A. Bridging Implicit andExplicit Solvent Approaches for Membrane Electrostatics. Biophys. J.2002, 83, 1374−1379.(45) Stern, H. A.; Feller, S. E. Calculation of the DielectricPermittivity Profile for a Nonuniform System: Application to a LipidBilayer Simulation. J. Chem. Phys. 2003, 118, 3401−3412.(46) Wander, M. C. F.; Clark, A. E. Structural and DielectricProperties of Quartz-Water Interfaces. J. Phys. Chem. C 2008, 112,19986−19994.(47) Frenkel, D.; Smit, B. Understanding Molecular Simulation: FromAlgorithms to Applications; Academic Press: London, 2002.(48) Kirkwood, J. G. The Dielectric Polarization of Polar Liquids. J.Chem. Phys. 1939, 7, 911−919.(49) Ballenegger, V.; Hansen, J.-P. Dielectric Permittivity Profiles ofConfined Polar Fluids. J. Chem. Phys. 2005, 122, 114711 1−10.(50) Bonthuis, D. J.; Gekle, S.; Netz, R. R. Dielectric Profile ofInterfacial Water and its Effect on Double-Layer Capacitance. Phys.Rev. Lett. 2011, 107, 166102 1−5.(51) Jackson, J. D. Classical Electrodynamics; John Wiley and Sons:New York, 1998.(52) Faraudo, J. The Missing Link Between the Hydration Force andInterfacial Water: Evidence from Computer Simulations. Curr. Opin.Colloid Interface Sci. 2011, 16, 557−560.(53) Yeh, I.; Berkowitz, M. L. Dielectric Constant of Water at HighElectric Fields: Molecular Dynamics Study. J. Chem. Phys. 1999, 110,7935−7942.(54) Abascal, J. L. F.; Vega, C. A General Purpose Model for theCondensed Phases of Water: TIP4P/2005. J. Chem. Phys. 2005, 123,234505 1−12.(55) Vega, C.; Abascal, J. L. F. Simulating Water with Rigid Non-Polarizable Models: A General Perspective. Phys. Chem. Chem. Phys.2011, 13, 19663−19688.(56) Steele, W. A. The Interaction of Gases with Solid Surfaces;Pergamon Press: Oxford, 1974.(57) Bandura, A. V.; Kubicki, J. D. Derivation of Force FieldParameters for TiO2-H2O Systems from ab Initio Calculations. J. Phys.Chem. B 2003, 107, 11072−11081.(58) Yeh, I.; Berkowitz, M. L. Ewald Summation for Systems withSlab Geometry. J. Chem. Phys. 1999, 111, 3155−3162.(59) Nose, S. A Unified Formulation of the Constant TemperatureMolecular Dynamics Methods. J. Chem. Phys. 1984, 81, 511−519.(60) Hoover, W. G. Canonical Dynamics: Equilibrium Phase-SpaceDistributions. Phys. Rev. A 1985, 31, 1695−1697.(61) Flyvbjerg, H.; Petersen, H. G. Error Estimates on Averages ofCorrelated Data. J. Chem. Phys. 1989, 91, 461−466.(62) Frohlich, H. Theory of Dielectrics; Clarendon Press: Oxford,1949.

(63) Predota, M.; Nezbeda, I.; Parez, S. Coarse-Grained Potential forInteraction with a Spherical Colloidal Particle and Planar Wall. Collect.Czech. Chem. Commun. 2010, 75, 527−545.(64) Predota, M.; Machesky, M. L.; Wesolowski, D. J.; Cummings, P.T. Electric Double Layer at the Rutile (110) Surface. 4. Effect ofTemperature and pH on the Adsorption and Dynamics of Ions. J. Phys.Chem. C 2013, 117, 22852−22866.(65) Ridley, M. K.; Hiemstra, T.; van Riemsdijk, W. H.; Machesky,M. L. Inner-Sphere Complexation of Cations at the Rutile−WaterInterface: A Concise Surface Structural Interpretation with the CD andMUSIC Model. Geochim. Cosmochim. Acta 2009, 73, 1841−1856.(66) Booth, F. The Dielectric Constant of Water and the SaturationEffect. J. Chem. Phys. 1951, 19, 391−394.(67) Aguilella-Arzo, M.; Andrio, A.; Aguilella, V. M.; Alcaraz, A.Dielectric Saturation of Water in a Membrane Protein Channel. Phys.Chem. Chem. Phys. 2009, 11, 358−365.

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp4128012 | J. Phys. Chem. C 2014, 118, 4818−48344834